Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.3 - Partial Derivatives - Exercises 14.3 - Page 807: 48


$w_{xx}=4x^2ye^{x^2-y}+2ye^{x^2-y}$ $w_{yy}=y^2e^{x^2-y}-2e^{x^2-y}$ $w_{xy}=w_{yx}=-2xye^{x^2-y}+2xe^{x^2-y}$

Work Step by Step

Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat y as a constant, and vice versa: $w_x=2xye^{x^2-y}$ $w_y=-ye^{x^2-y}+e^{x^2-y}$ Then take the derivative of the first order partialderivatives to find second partial derivatives: $w_{xx}=4x^2ye^{x^2-y}+2ye^{x^2-y}$ $w_{yy}=y^2e^{x^2-y}-e^{x^2-y}-e^{x^2-y}=y^2e^{x^2-y}-2e^{x^2-y}$ Second partial derivatives of first order partial derivative of x with respect to y and y with respect to x are the same: $w_{xy}=w_{yx}=-2xye^{x^2-y}+2xe^{x^2-y}$
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